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<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
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<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
<li><a href="sec_2-intro.html" data-scroll="sec_2-intro" class="internal">Linear and Nonlinear Equation</a></li>
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<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
<li><a href="sec2_1.html" data-scroll="sec2_1" class="internal">Linear Equations</a></li>
<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
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<li><a href="sec3_1.html" data-scroll="sec3_1" class="active">Homogeneous equations with constant coefficient</a></li>
<li><a href="sec3_2.html" data-scroll="sec3_2" class="internal">Fundamental Solutions of Linear Homogeneous Equations</a></li>
<li><a href="sec3_3.html" data-scroll="sec3_3" class="internal">Linear Independence and Wronskian</a></li>
<li><a href="sec3_4.html" data-scroll="sec3_4" class="internal">Complex roots of the characteristic equations</a></li>
<li><a href="sec3_5.html" data-scroll="sec3_5" class="internal">Repeated Roots: Reduction of Order</a></li>
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<li><a href="sec4_1.html" data-scroll="sec4_1" class="internal">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</a></li>
<li><a href="sec4_2.html" data-scroll="sec4_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec4_3.html" data-scroll="sec4_3" class="internal">The Method of Undetermined Coefficients</a></li>
<li><a href="sec4_4.html" data-scroll="sec4_4" class="internal">The Method of Variation of Parameters</a></li>
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<a href="ch_five.html" data-scroll="ch_five" class="internal"><span class="codenumber">5</span> <span class="title">Series Solutions of Second Order Linear Equations</span></a><ul>
<li><a href="sec5_1.html" data-scroll="sec5_1" class="internal">Brief Review on Power Series</a></li>
<li><a href="sec5_2.html" data-scroll="sec5_2" class="internal">Introduction</a></li>
<li><a href="sec5_3.html" data-scroll="sec5_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec5_4.html" data-scroll="sec5_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec5_5.html" data-scroll="sec5_5" class="internal">Series Solution near a Regular Singular Point</a></li>
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<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec6_1.html" data-scroll="sec6_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec6_2.html" data-scroll="sec6_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
<li><a href="sec6_3.html" data-scroll="sec6_3" class="internal">Complex Eigenvalues</a></li>
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<li><a href="sec6_6.html" data-scroll="sec6_6" class="internal">Non-homogeneous linear systems</a></li>
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<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec7_1.html" data-scroll="sec7_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec7_2.html" data-scroll="sec7_2" class="internal">Eigenvalue Problems</a></li>
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<li><a href="sec7_4.html" data-scroll="sec7_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec7_5.html" data-scroll="sec7_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec7_6.html" data-scroll="sec7_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec7_7.html" data-scroll="sec7_7" class="internal">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec7_8.html" data-scroll="sec7_8" class="internal">Other Heat Conduction Problems</a></li>
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<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
<li><a href="sec8_2.html" data-scroll="sec8_2" class="internal">Finding Laplace Transforms</a></li>
<li><a href="sec8_3.html" data-scroll="sec8_3" class="internal">Finding inverse transforms using partial fractions</a></li>
<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="internal">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="internal">Laplace transform for PDE (heat equation)</a></li>
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<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
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<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec3_1"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">3.1</span> <span class="title">Homogeneous equations with constant coefficient</span>
</h2>
<p id="p-58">The <dfn class="terminology">general form of second order ODE</dfn> is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_1.html ./knowl/eq3_2.html">
\begin{equation*}
\frac{\textrm{d}^2 y}{\textrm{d} x^2}=f(x, y, \frac{\textrm{d} y}{\textrm{d} x}).
\end{equation*}
</div>
<p class="continuation">In particular, we consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_1.html ./knowl/eq3_2.html">
\begin{equation*}
f(x, y,  \frac{\textrm{d} y}{\textrm{d} x})=-p(x)  \frac{\textrm{d} y}{\textrm{d} x}-q(x) y+g(x),
\end{equation*}
</div>
<p class="continuation">i. e.,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_1.html ./knowl/eq3_2.html" id="eq3_1">
\begin{equation}
\frac{\textrm{d}^2 y}{\textrm{d} x^2}+p(x) \frac{\textrm{d} y}{\textrm{d} x}+q(x) y=g(x).\tag{3.1.1}
\end{equation}
</div>
<p class="continuation">This is the <dfn class="terminology">general form of second order linear ODE</dfn>. Initial conditions</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_1.html ./knowl/eq3_2.html" id="eq3_2">
\begin{equation}
y(x_0)=y_0,\quad y^{\prime}(x_0)=y_1.\tag{3.1.2}
\end{equation}
</div>
<p class="continuation">Equations (<a href="" class="xref" data-knowl="./knowl/eq3_1.html" title="Equation 3.1.1">(3.1.1)</a>) together with (<a href="" class="xref" data-knowl="./knowl/eq3_2.html" title="Equation 3.1.2">(3.1.2)</a>) are called an <dfn class="terminology">initial value problem</dfn>.</p>
<p id="p-59">If <span class="process-math">\(g(x)=0\text{,}\)</span> (3) is called <dfn class="terminology">homogeneous</dfn>, otherwise, it is <dfn class="terminology">non-homogeneous</dfn>. In this section, we consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
p(x)=b,\quad q(x)=c,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(b, c\)</span> are constants, i. e.,</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="eq3_3">
\begin{equation}
y^{\prime \prime}+b y^{\prime}+c y=0.\tag{3.1.3}
\end{equation}
</div>
<p id="p-60">First, we consider a particular example.</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="eq3_3_1">
\begin{equation}
y^{\prime \prime}-y=0.\tag{3.1.4}
\end{equation}
</div>
<p class="continuation">Inspect: <span class="process-math">\(y^{\prime \prime}=y\)</span> and note that <span class="process-math">\(((e)^x)^{\prime}=e^x\text{,}\)</span> <span class="process-math">\((e^x)^{\prime \prime}=e^x\text{.}\)</span> Thus, <span class="process-math">\(y_1(x)=e^x\)</span> is a solution. Also, <span class="process-math">\((e^{-x})^{\prime \prime}=e^{-x}\text{.}\)</span> Thus, <span class="process-math">\(y_2(x)=e^{-x}\)</span> is another solution. Further, <span class="process-math">\(C_1 e^x\)</span> and <span class="process-math">\(C_2 e^{-x}\)</span> (<span class="process-math">\(C_1\)</span> and <span class="process-math">\(C_2\)</span> are any constants) are still two solutions. Also,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
(C_1 e^x+C_2 e^{-x})^{\prime \prime}=C_1 e^x+C_2 e^{-x}.
\end{equation*}
</div>
<p class="continuation">Thus, <span class="process-math">\(C_1 e^x+C_2 e^{-x}\)</span> is also the solution, representing double infinite family of solutions, see Figure <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "fig3" missing or not unique]</code>.</p>
<p id="p-61">Suppose that we further impose</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="eq3_4_1">
\begin{equation}
y(0)=2,\quad y^{\prime}(0)=-1.\tag{3.1.5}
\end{equation}
</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_3_1.html ./knowl/eq3_4_1.html" id="p-62">
\begin{equation*}
\begin{aligned}
&amp;y(0)=2:\quad 2=C_1+C_2,\\
&amp; y^{\prime}(0)=-1:\quad -1=C_1-C_2,
&amp;~\rightarrow~C_1=\frac{1}{2},\quad C_2=\frac{3}{2}.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Thus,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_3_1.html ./knowl/eq3_4_1.html">
\begin{equation*}
y=\frac{1}{2} e^x+\frac{3}{2} e^{-x}.
\end{equation*}
</div>
<p class="continuation">This is the solution to the initial value problem of (<a href="" class="xref" data-knowl="./knowl/eq3_3_1.html" title="Equation 3.1.4">(3.1.4)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq3_4_1.html" title="Equation 3.1.5">(3.1.5)</a>).</p>
<p id="p-63">Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_4.html ./knowl/eq3_5.html ./knowl/eq3_5.html" id="eq3_5">
\begin{equation}
y^{\prime \prime}+b y^{\prime}+c y=0.\tag{3.1.6}
\end{equation}
</div>
<p class="continuation">Seek a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_4.html ./knowl/eq3_5.html ./knowl/eq3_5.html" id="eq3_4">
\begin{equation}
y=e^{rx},\tag{3.1.7}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\(r\)</span> needs to be determined. Substituting (<a href="" class="xref" data-knowl="./knowl/eq3_4.html" title="Equation 3.1.7">(3.1.7)</a>) into (<a href="" class="xref" data-knowl="./knowl/eq3_5.html" title="Equation 3.1.6">(3.1.6)</a>):</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_4.html ./knowl/eq3_5.html ./knowl/eq3_5.html">
\begin{equation*}
r^2 e^{rx}+bre^{rx}+ce^{rx}=0~\rightarrow~r^2+br+c=0.
\end{equation*}
</div>
<p class="continuation">This algebraic equation for <span class="process-math">\(r\)</span> is called the <dfn class="terminology">characteristic equation</dfn> of (<a href="" class="xref" data-knowl="./knowl/eq3_5.html" title="Equation 3.1.6">(3.1.6)</a>).</p>
<p id="p-64">For the solution of the characteristic equation, there are three cases:</p>
<p id="p-65">(Case a) Two distinct real roots: <span class="process-math">\(r_1\)</span> and <span class="process-math">\(r_2\text{.}\)</span>(Case b) Two repeated roots: <span class="process-math">\(r_1=r_2\text{.}\)</span>(Case c) Two complex conjugate roots: <span class="process-math">\(r_1, r_2\)</span> where <span class="process-math">\(r_2=\overline{r_1}\text{.}\)</span></p>
<p id="p-66">In this section, we only consider (Case a). Thus, we have two solutions:</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_5.html ./knowl/eq3_5_3.html">
\begin{equation*}
y_1(x)=e^{r_1 x},\quad y_2(x)=e^{r_2 x}.
\end{equation*}
</div>
<p class="continuation">It can be verified that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_5.html ./knowl/eq3_5_3.html" id="eq3_5_3">
\begin{equation}
y(x)=C_1 e^{r_1 x}+C_2 e^{r_2 x},\tag{3.1.8}
\end{equation}
</div>
<p class="continuation">satisfies the (<a href="" class="xref" data-knowl="./knowl/eq3_5.html" title="Equation 3.1.6">(3.1.6)</a>) and thus is also a solution. Here, <span class="process-math">\(C_1, C_2\)</span> are arbitrary constants. It turns out that (<a href="" class="xref" data-knowl="./knowl/eq3_5_3.html" title="Equation 3.1.8">(3.1.8)</a>) contains every possible solutions, i.e., it is the <dfn class="terminology">general solution</dfn> (this will be justified later on).</p>
<p id="p-67">Next, suppose that we impose the initial conditions</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_5.html ./knowl/eq3_5_2.html" id="eq3_5_2">
\begin{equation}
y(x_0)=y_0,\quad y^{\prime}(x_0)=y_1.\tag{3.1.9}
\end{equation}
</div>
<p class="continuation">Then we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_5.html ./knowl/eq3_5_2.html">
\begin{equation*}
\begin{aligned}
&amp;y(x_0)=y_0: ~ y_0=C_1 e^{r_1 x_0}+C_2 e^{r_2 x_0};\\
&amp; y^{\prime}(x_0)=y_1: ~ y_1=C_1 r_1 e^{r_1 x_0}+C_2 r_2 e^{r_2 x_0}.\\
&amp;\rightarrow C_1=\frac{y_1-y_0 r_2}{r_1-r_2} e^{-r_1 x_0},\quad C_2=\frac{y_0 r_1-y_1}{r_1-r_2} e^{-r_2 x_0}.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Since <span class="process-math">\(C_1\)</span> and <span class="process-math">\(C_2\)</span> are uniquely determined, we have the unique solution to the initial value problem (<a href="" class="xref" data-knowl="./knowl/eq3_5.html" title="Equation 3.1.6">(3.1.6)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq3_5_2.html" title="Equation 3.1.9">(3.1.9)</a>), which is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_5.html ./knowl/eq3_5_2.html">
\begin{equation*}
y=\frac{y_1-y_0 r_2}{r_1-r_2} e^{-r_1 x_0} e^{r_1 x}+\frac{y_0 r_1-y_1}{r_1-r_2} e^{-r_2 x_0} e^{r_2 x}.
\end{equation*}
</div>
<p id="p-68"><dfn class="terminology">Example 1:</dfn> Find the general solution of the following ODE:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
2 y^{\prime \prime}-3 y^{\prime}+y=0.
\end{equation*}
</div>
<p id="p-69"><dfn class="terminology">Solution:</dfn> Seek a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y=e^{rx}.
\end{equation*}
</div>
<p class="continuation">Substituting it into the ODE,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
2 r^2 e^{rx}-3 r e^{rx}+e^{rx}=0 ~\rightarrow~2 r^2-3r+1=0~\rightarrow~(2r-1)(r-1)=0~\rightarrow~r=\frac{1}{2}~ \textrm{or} ~1.
\end{equation*}
</div>
<p class="continuation">Thus, there are two solutions:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y_1=e^{\frac{1}{2}x},\quad y_2=e^x.
\end{equation*}
</div>
<p class="continuation">The general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y=C_1 e^{\frac{1}{2}x}+C_2 e^x,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(C_1\)</span> and <span class="process-math">\(C_2\)</span> are arbitrary constants.</p></section></div></main>
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